Final answer:
A possible length for the third side of the triangle, according to the Triangle Angle Bisector Theorem and given the lengths of the other sides and segments, is 8.8 cm.
Step-by-step explanation:
The Triangle Angle Bisector Theorem states that the angle bisector of a triangle divides the opposite side into two segments that are proportional to the lengths of the other two sides of the triangle. In this question, we have segments of lengths 8 cm and 4 cm, which means the ratio of the other two sides must also be 2:1, since 8 cm is double 4 cm.
Given one of the sides of the triangle is 4.4 cm, it corresponds to either the longer or shorter segment. If the 4.4 cm side is opposite the shorter segment (4 cm), then using the ratio 2:1, the length of the third side should be twice that of the given side, which would be 8.8 cm. However, if the 4.4 cm side is opposite the longer segment (8 cm), the third side should be half of the given side, which is not possible since the lengths of the sides of a triangle must be positive.
Therefore, a possible length for the third side, based on the Triangle Angle Bisector Theorem, is 8.8 cm.